theorem :: FDIFF_12:49

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_left_differentiable_in x0 & f2 is_right_differentiable_in f1 . x0 & ex r being Real st
( r > 0 & f1 | [.(x0 - r),x0.] is non-increasing ) holds
( f2 * f1 is_left_differentiable_in x0 & Ldiff ((f2 * f1),x0) = (Rdiff (f2,(f1 . x0))) * (Ldiff (f1,x0)) )